Kinetic energy and momentum.

Ek = 0.5mv^2
P=mv

m=1, v=1
Ek = 0.5

m=1, v=2
Ek = 2

Accelerating a 1 kg object to 1m/s requires 0.5 joules whilst accelerating it a further 1m/s to 2m/s requires an additional 1.5 joules. Why is this?

>>4
This is because when you accelerate something over a certain distance, the acceleration*distance is directly proportional to the differences of the SQUARES of the velocities:

b^2=a^2+2*acceleration*distance

The reason why the above equation is so is simply by considering the fact that for any function f such that f''(x)=k where k is a constant, f'(b)^2-f'(a)^2=(f'(a)+(b-a)k)^2-f'(a)^2=f'(a)^2-f'(a)^2+2f'(a)(b-a)k+(k^2)(b-a)^2=2f'(a)bk-2f'(a)ak+(k^2)(b-a)^2=2k(f'(a)(b-a)+(1/2)k(b-a)^2)=2k(f(b)-f(a)).